NCERT Solutions for Class 12 Maths Chapter 5 – Continuity and Differentiability Ex 5.4
Page No 174:
Question 1:
Differentiate the following w.r.t. x:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6897/Item%2060_html_5a6ef49c.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6897/Item%2060_html_2a0559ed.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6897/Item%2060_html_2a0559ed.gif)
By using the quotient rule, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6897/Item%2060_html_m44826895.gif)
Question 2:
Differentiate the following w.r.t. x:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6899/Chapter%205_html_10665804.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6899/Chapter%205_html_m69c3638f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6899/Chapter%205_html_m69c3638f.gif)
By using the chain rule, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6899/Chapter%205_html_m61ef7eca.gif)
Question 3:
Differentiate the following w.r.t. x:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6902/Chapter%205_html_62b1ecd9.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6902/Chapter%205_html_41e9a261.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6902/Chapter%205_html_41e9a261.gif)
By using the chain rule, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6902/Chapter%205_html_m6cd6d4fc.gif)
Question 4:
Differentiate the following w.r.t. x:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6904/Chapter%205_html_m417e6043.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6904/Chapter%205_html_m85aa204.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6904/Chapter%205_html_m85aa204.gif)
By using the chain rule, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6904/Chapter%205_html_m2beb318a.gif)
Question 5:
Differentiate the following w.r.t. x:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6906/Chapter%205_html_67353335.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6906/Chapter%205_html_m68f88f2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6906/Chapter%205_html_m68f88f2.gif)
By using the chain rule, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6906/Chapter%205_html_f778309.gif)
Question 6:
Differentiate the following w.r.t. x:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6907/Chapter%205_html_7cef6427.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6907/Chapter%205_html_m6e457112.gif)
Question 7:
Differentiate the following w.r.t. x:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6908/Chapter%205_html_m289c9b29.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6908/Chapter%205_html_mb6637ee.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6908/Chapter%205_html_mb6637ee.gif)
Then, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6908/Chapter%205_html_43111adc.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6908/Chapter%205_html_43111adc.gif)
By differentiating this relationship with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6908/Chapter%205_html_63fdcfa6.gif)
Question 8:
Differentiate the following w.r.t. x:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6910/Chapter%205_html_5f9334b3.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6910/Chapter%205_html_6451572a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6910/Chapter%205_html_6451572a.gif)
By using the chain rule, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6910/Chapter%205_html_m72ced0c9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6910/Chapter%205_html_56a4a2d6.gif)
Question 9:
Differentiate the following w.r.t. x:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6912/Chapter%205_html_m1401ed6e.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6912/Chapter%205_html_4314d40d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6912/Chapter%205_html_4314d40d.gif)
By using the quotient rule, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6912/Chapter%205_html_6256b73f.gif)
Question 10:
Differentiate the following w.r.t. x:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6913/Chapter%205_html_a5d499d.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6913/Chapter%205_html_26fadfea.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6913/Chapter%205_html_26fadfea.gif)
By using the chain rule, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6913/Chapter%205_html_m7e614266.gif)